Question: Determine how many solutions exist for the system of equations. ${5x+y = 5}$ ${-2x+y = -2}$
Answer: Convert both equations to slope-intercept form: ${5x+y = 5}$ $5x{-5x} + y = 5{-5x}$ $y = 5-5x$ ${y = -5x+5}$ ${-2x+y = -2}$ $-2x{+2x} + y = -2{+2x}$ $y = -2+2x$ ${y = 2x-2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x+5}$ ${y = 2x-2}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.